Geometric representation and the adiabatic geometric phase in four-wave mixing processes.

The application of the adiabatic geometric phase (AGP) to nonlinear frequency conversion may help to develop new types of all-optical devices, which leads to all-optical modulation of the phase front of one wave by the intensity of other waves. In this paper, we develop the canonical Hamilton equation and a corresponding geometric representation for two schemes of four-wave mixing (FWM) processes (ω + ω = ω + ω and ω + ω + ω = ω), which can precisely describe and calculate the AGP controlled by the quasi-phase matching technique. The AGPs of the idler (ω) and signal (ω) waves for these two schemes of FWM are studied systematically when the two pump waves (ω and ω) are in either the undepleted or in the depleted pump cases, respectively. The analysis reveals that the proposed methods for calculating the AGP are universal in both cases. We expect that the analysis of AGP in FWM processes can be applied to all-optically shaping or encoding of ultrafast light pulse.